1
From Atkins and Paula, Physical Chemistry
Chapter 18, Molecular Interactions
(Material in sec.18.4 is the most important section; this is how
macromolecules and aggregates stay together, and why things are sticky!)
18.1 Electric dipole moments
Polar molecules
Polarization
18.2 Polarizabilities (omitted--used in a couple of sections of 18.4; don’t
worry about it)
18.3 Relative permittivities (just skim--18.3 is only important to see how
electrostatic interactions are “muted” in water or other substances.)
18.4. Interactions between dipoles (8 parts)
Follow the math on the subsections with stars only if you think you have the
background--it might be rewarding for future use; read the rest in detail.
*a. The potential energy of interaction, p.13
*b. The electric field, p.18
*c. Dipole-dipole interactions (Keesom interaction, p. 19)
d. Dipole-induced-dipole interactions (p.24)
e. Induced-dipole-induced-dipole interactions (dispersion or London
forces, p.26) [I may give you longer derivation and discussion of this.]
f. Hydrogen bonding [p.28. I definitely will give you a separate reading
oriented toward biological macromolecules and water.]
g. Hydrophobic interaction [ p.30. Important for lipid bilayer structure of
cell membrane as well as many properties of proteins (e.g. notice which
amino acids are hydrophobic, hydrophilic), etc.]
h. Total attractive interactions (p.31)
18.5. Repulsive and total interactions (p.31)
Molecular recognition and drug design (I 18.1, p.35)
Questions and exercises (p.39).
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Molecular interactions are responsible for the unique properties of
substances as simple as water and as complex as polymers. We begin our
examination of molecular interactions by describing the electric properties of
molecules, which may be interpreted in terms of concepts in electronic
structure. We shall see that small imbalances of charge distributions in
molecules allow them to interact with one another and with externally
applied fields. One result of this interaction is the cohesion of molecules to
form the bulk phases of matter. These noncovalent molecular
interactions are also important for understanding the shapes adopted by
biological and synthetic macromolecules, as we shall see in Chapter 19. (I
will distribute an edited version of that chapter, applying (easy) equilibrium
statistical mechanics to biological polymers, separately).
Electric properties of molecules
Many of the electric properties of molecules can be traced to the competing
influences of nuclei with different charges or the competition between the
control exercised by a nucleus and the influence of an externally applied
field. The former competition may result in an electric dipole moment. The
latter may result in properties such as refractive index and optical activity.
[We don’t care about the latter, only that you can induce a dipole moment,
for the calculation of London forces.]
3
18.
1
Electric dipole moments
An electric dipole consists of two electric charges +q and _q separated by
a distance R.
This arrangement of charges is represented by a
vector m (1). The magnitude of m is µ = qR and,
although the SI unit of dipole moment is coulomb
metre (C m), it is still commonly reported in the nonSI unit debye, D, named after Peter Debye, a pioneer
in the study of dipole moments of molecules, where
(18.1)
The dipole moment of a pair of charges +e and _e separated by 100 pm is
_29
1.6 _ 10
C m, corresponding to 4.8 D. Dipole moments of small molecules
are typically about 1 D. The conversion factor in eqn 18.1 stems from the
original definition of the debye in terms of c.g.s. units: 1 D is the dipole
moment of two equal and opposite charges of magnitude 1 e.s.u. separated
by 1 Å.
(a) Polar molecules
A polar molecule is a molecule with a
permanent electric dipole moment. The
permanent dipole moment stems from
the partial charges on the atoms in the
molecule that arise from differences in
electronegativity or other features of
bonding (Section 11-6). Nonpolar
molecules acquire an induced dipole
moment in an electric field on account of
the distortion the field causes in their
electronic distributions and nuclear
positions; however, this induced moment
is only temporary, and disappears as
soon as the perturbing field is removed.
Polar molecules also have their existing dipole moments temporarily
modified by an applied field.
In elementary chemistry, an electric dipole moment is represented by the
arrow added to the Lewis structure for the molecule, with the + marking the
4
positive end. Note that the direction of the arrow is opposite to that of µ.
The Stark effect (Section 13-5) is used to measure
the electric dipole moments of molecules for which a
rotational spectrum can be observed. In many cases
microwave spectroscopy cannot be used because the
sample is not volatile, decomposes on vaporization, or
consists of molecules that are so complex that their
rotational spectra cannot be interpreted. In such cases
the dipole moment may be obtained by measurements
on a liquid or solid bulk sample using a method
explained later. Computational software is now widely
available, and typically computes electric dipole
moments by assessing the electron density at each
point in the molecule and its coordinates relative to the
centroid of the molecule; however, it is still important
to be able to formulate simple models of the origin of these moments and to
understand how they arise. The following paragraphs focus on this aspect.
All heteronuclear diatomic molecules are polar, and typical values of µ
include 1.08 D for HCl and 0.42 D for HI (Table 18-1). Molecular symmetry
is of the greatest importance in deciding whether a polyatomic molecule is
polar or not. Indeed, molecular symmetry is more important than the
question of whether or not the atoms in the molecule belong to the same
element. Homonuclear polyatomic molecules may be polar if they have low
symmetry and the atoms are in inequivalent positions. For instance, the
angular molecule ozone, O3 (2), is homonuclear; however, it is polar
because the central O atom is different from the outer two (it is bonded to
two atoms, they are bonded only to one); moreover, the dipole moments
associated with each bond make an angle to each other and do not cancel.
Heteronuclear polyatomic molecules may be nonpolar if they have high
symmetry, because individual bond dipoles may then cancel. The
heteronuclear linear triatomic molecule CO2, for example, is nonpolar
because, although there are partial charges on all three atoms, the dipole
moment associated with the OC bond points in the opposite direction to the
dipole moment associated with the CO bond, and the two cancel (3).
To a first approximation, it is possible to resolve the dipole moment of a
polyatomic molecule into contributions from various groups of atoms in the
molecule and the directions in which these individual contributions lie
(Fig. 18.1).
5
Fig. 18.1 The resultant dipole moments (pale yellow) of
the dichlorobenzene isomers (b to d) can be obtained
approximately by vectorial addition of two chlorobenzene
dipole moments (1.57 D), purple.
Thus, 1,4-dichlorobenzene is nonpolar by symmetry on
account of the cancellation of two equal but opposing
C—Cl moments (exactly as in carbon dioxide). 1,2Dichlorobenzene, however, has a dipole moment which is
approximately the resultant of two chlorobenzene dipole
moments arranged at 60° to each other. This technique of
‘vector addition’ can be applied with fair success to other
series of related molecules, and the resultant µres of two
dipole moments µ1 and µ2 that make an angle _ to each
other (4)
is approximately
(18.2a)
When the two dipole moments have the same magnitude (as in the
dichlorobenzenes), this equation simplifies to
(18.2b)
Self Test 18.1 Estimate the ratio of the electric dipole moments of ortho
(1,2-) and meta (1,3-) disubstituted benzenes.
Correct Answer
µ(ortho)/µ(meta) = 1.7
A better approach to the calculation of dipole moments is to take into
account the locations and magnitudes of the partial charges on all the
atoms. These partial charges are included in the output of many molecular
structure software packages. To calculate the x-component, for instance, we
need to know the partial charge on each atom and the atom’s x-coordinate
relative to a point in the molecule and form the sum
6
(18.3a)
Here qJ is the partial charge of atom J,xJ is the x-coordinate of atom J, and
the sum is over all the atoms in the molecule. Analogous expressions are
used for the y- and z-components. For an electrically neutral molecule, the
origin of the coordinates is arbitrary, so it is best chosen to simplify the
measurements. In common with all vectors, the magnitude of m is related to
the three components µx,µy, and µz by
(18.3b)
Example 18.1 Calculating a molecular dipole moment
Estimate the electric dipole moment of the amide group shown in (5) by using
the partial charges (as multiples of e) in Table 18-2
and the locations of the atoms shown.
Method We use eqn 18.3a to calculate each of the components of the dipole
moment and then eqn 18.3b to assemble the three components into the
magnitude of the dipole moment. Note that the partial charges are multiples of
the fundamental charge, e = 1.609 _ 10
Answer The expression for µx is
_19
C.
7
corresponding to µx = 0.42 D. The expression for µy is:
It follows that µy = _2.7 D. Therefore,because µz = 0,
We can find the orientation of the dipole moment by
arranging an arrow of length 2.7 units of length to
have x, y, and z components of 0.42, _2.7, and 0 units;
the orientation is superimposed on (6).
Self Test 18.2 Calculate the electric dipole moment of
formaldehyde, using the information in (7).
Correct Answer: - 3.2D.
(b) Polarization
The polarization, P, of a sample is the electric
dipole moment density, the mean electric dipole
moment of the molecules, µ, multiplied by the
number density, :
(18.4)
In the following pages we refer to the sample as a dielectric, by which is
meant a polarizable, nonconducting medium.
The polarization of an isotropic fluid sample is zero in the absence of an
applied field because the molecules adopt random orientations, so µ = 0. In
the presence of a field, the dipoles become partially aligned because some
orientations have lower energies than others. As a result, the electric dipole
moment density is nonzero. We show in the Justification below that, at a
temperature T
(18.5)
where z is the direction of the applied field . Moreover, as we shall see, there
8
is an additional contribution from the dipole moment induced by the field.
Justification 18.1 The thermally averaged dipole moment
The probability dp that a dipole has an orientation in the range _ to _ + d_ is
given by the Boltzmann distribution (Section 16-1b), which in this case is
where E(_) is the energy of the dipole in the field: E(_) = _µ cos _, with 0 ≤
_ ≤ ". The average value of the component of the dipole moment parallel to
the applied electric field is therefore
with x = µ/kT. The integral takes on a simpler appearance when we write y
= cos _ and note that dy = _sin _ d_.
At this point we use
It is now straightforward algebra to combine these two results and to obtain
9
(18.6)
L(x) is called the Langevin function.
Under most circumstances, x is very small (for example, if µ = 1 D and T =
_1
300 K, then x exceeds 0.01 only if the field strength exceeds 100 kV cm ,
and most measurements are done at much lower strengths). When x 1, the
exponentials in the Langevin function can be expanded, and the largest term
that survives is
When x is small, it is possible to simplify expressions by using the expansion
x
1
2
1
3
e = 1 + x + –2x + –6x + · · · ; it is important when developing
approximations that all terms of the same order are retained because loworder terms might cancel.
(18.7)
Therefore, the average molecular dipole moment is given by eqn 18.6.
10
18-3
Relative permittivities
When two charges q1 and q2 are separated by a distance r in a vacuum, the
potential energy of their interaction is (see Appendix 3):
(18.12a)
When the same two charges are immersed in a medium (such as air or a
liquid), their potential energy is reduced to
(18.12b)
where _ is the permittivity of the medium. The permittivity is normally
expressed in terms of the dimensionless relative permittivity, _r, (formerly
and still widely called the dielectric constant) of the medium:
(18.13)
The relative permittivity can have a very significant effect on the strength of
the interactions between ions in solution. For instance, water has a relative
permittivity of 78 at 25°C, so the interionic Coulombic interaction energy is
reduced by nearly two orders of magnitude from its vacuum value. Some of
the consequences of this reduction for electrolyte solutions were explored in
Chapter 5.
The relative permittivity of a substance is large if its molecules are polar or
highly polarizable. The quantitative relation between the relative permittivity
and the electric properties of the molecules is obtained by considering the
polarization of a medium, and is expressed by the Debye equation (for the
derivation of this and the following equations, see Further reading):
(18.14)
where _ is the mass density of the sample, M is the molar mass of the
molecules, and Pm is the molar polarization, which is defined as
(18.15)
2
The term µ /3kT stems from the thermal averaging of the electric dipole
moment in the presence of the applied field (eqn 18.5). The corresponding
11
expression without the contribution from the permanent dipole moment is
called the Clausius–Mossotti equation:
(18.16)
The Clausius–Mossotti equation is used when there is no contribution from
permanent electric dipole moments to the polarization, either because the
molecules are nonpolar or because the frequency of the applied field is so
high that the molecules cannot orientate quickly enough to follow the change
in direction of the field.
Example 18.2 Determining dipole moment and polarizability
The relative permittivity of a substance is measured by comparing the
capacitance of a capacitor with and without the sample present (C and C0,
respectively) and using _r = C/C0. The relative permittivity of camphor (8)
was measured at a series of temperatures with the results given below.
Determine the dipole moment and the polarizability volume of the molecule.
Method Equation 18.14 implies that the polarizability and permanent
electric dipole moment of the molecules in a sample can be determined by
measuring _r at a series of temperatures, calculating Pm, and plotting it
2
against 1/T. The slope of the graph is NAµ /9_0k and its intercept at 1/T = 0
is NA_/3_0. We need to calculate (_r _ 1)/(_r + 2) at each temperature, and
12
then multiply by M/_ to form Pm.
_1
Answer For camphor, M = 152.23 g mol . We can therefore use the data to
draw up the following table:
The points are plotted in Fig. 18.2. The
intercept lies at 82.7, so __ = 3.3 _ 10
_23
_30
3
cm .
The slope is 10.9, so µ = 4.46 _ 10
C m,
corresponding to 1.34 D. Because the Debye
equation describes molecules that are free to
rotate, the data show that camphor, which does
not melt until 175°C, is rotating even in the
solid. It is an approximately spherical molecule
The Maxwell equations that describe the
properties of electromagnetic radiation (see
Further reading) relate the refractive index at
a (visible or ultraviolet) specified wavelength to
the relative permittivity at that frequency:
(18.17)
Therefore, the molar polarization, Pm, and the molecular polarizability, _, can
be measured at frequencies typical of visible light (about 10
measuring the refractive index of the sample and using the
Clausius–Mossotti equation.
15
to 10
16
Hz) by
13
Interactions between molecules
A van der Waals interaction is the attractive interaction between closed-
6
shell molecules that depends on the distance between the molecules as 1/r .
In addition, there are interactions between ions and the partial charges of
polar molecules and repulsive interactions that prevent the complete
collapse of matter to nuclear densities. The repulsive interactions arise from
Coulombic repulsions and, indirectly, from the Pauli principle and the
exclusion of electrons from regions of space where the orbitals of
neighbouring species overlap.
18-4
Interactions between dipoles (eight subsections)
Most of the discussion in this section is based on the Coulombic potential
energy of interaction between two charges (eqn 18.12a). We can easily
adapt this expression to find the potential energy of a point charge and a
dipole and to extend it to the interaction between two dipoles.
(a) The potential energy of interaction
We show in the Justification below that the potential energy of interaction
between a point dipole µ1 = q1l and the point charge q2 in the arrangement
shown in (9)
is
(18.18)
With µ in coulomb metres, q2 in coulombs, and r in metres, V is obtained in
joules. A point dipole is a dipole in which the separation between the
charges is much smaller than the distance at which the dipole is being
observed, l r. The potential energy rises towards zero (the value at infinite
2
separation of the charge and the dipole) more rapidly (as 1/r ) than that
between two point charges (which varies as 1/r) because, from the
viewpoint of the point charge, the partial charges of the dipole seem to
14
merge and cancel as the distance r increases (Fig. 18.3).
Fig. 18.3 There are two contributions to
the diminishing field of an electric dipole
with distance (here seen from the side).
The potentials of the charges decrease
(shown here by a fading intensity) and the
two charges appear to merge, so their
combined effect approaches zero more
rapidly than by the distance effect alone.
Justification 18.4 The interaction between a point
charge and a point dipole
The sum of the potential energies of repulsion between like charges and
attraction between opposite charges in the orientation shown in (9) is
where x = l/2r. Because l r for a point dipole, this expression can be
simplified by expanding the terms in x and retaining only the leading term:
With µ1 = q1l, this expression becomes eqn 18.18. This expression should
be multiplied by cos _ when the point charge lies at an angle _ to the axis of
the dipole.
............................................................
The following expansions are often useful:
Example 18.3 Calculating the interaction energy of two dipoles
Calculate the potential energy of interaction of two dipoles in the
arrangement shown in (10) when their separation is r.
15
Method We proceed in exactly the same way as in Justification 18.4, but
now the total interaction energy is the sum of four pairwise terms, two
attractions between opposite charges, which contribute negative terms to
the potential energy, and two repulsions between like charges, which
contribute positive terms.
Answer The sum of the four contributions is
with x = l/r. As before, provided l r we can expand the two terms in x and
2
retain only the first surviving term, which is equal to 2x . This step results in
the expression
Therefore, because µ1 = q1l and µ2 = q2l, the potential energy of interaction
in the alignment shown in the illustration is
3
This interaction energy approaches zero more rapidly (as 1/r ) than for the
previous case: now both interacting entities appear neutral to each other at
large separations. See Further information 18.1 for the general expression.
Self Test 18.4 Derive an expression for the potential energy when the
dipoles are in the arrangement shown in (11).
Correct Answer
V = µ1µ2/4"_0r3
Table 18-3 summarizes the various expressions for the interaction of
charges and dipoles.
16
It is quite easy to extend the formulas given
there to obtain expressions for the energy of
interaction of higher multipoles, or arrays of
point charges (Fig. 18.4).
Fig. 18.4 Typical charge arrays corresponding
to electric multipoles. The field arising from an
arbitrary finite charge distribution can be
expressed as the superposition of the fields
arising from a superposition of multipoles.
Specifically, an n-pole is an array of point
charges with an n-pole moment but no lower
moment. Thus, a monopole (n = 1) is a point
charge, and the monopole moment is what we
normally call the overall charge. A dipole (n =
2), as we have seen, is an array of charges that
has no monopole moment (no net charge). A
quadrupole (n = 3) consists of an array of
point charges that has neither net charge nor
dipole moment (as for CO2 molecules, 3). An
17
octupole (n = 4) consists of an array of point charges that sum to zero and
which has neither a dipole moment nor a quadrupole moment (as for CH4
molecules, 12). The feature to remember is that the interaction energy falls
off more rapidly the higher the order of the multipole. For the interaction of
an n-pole with an m-pole, the potential energy varies with distance as
(18.19)
Comment 18.8
The reason for the even steeper decrease with distance is the same as
before: the array of charges appears to blend together into neutrality more
rapidly with distance the higher the number of individual charges that
contribute to the multipole. Note that a given molecule may have a charge
distribution that corresponds to a superposition of several different
multipoles.
18
(b) The electric field
The same kind of argument as that used to derive expressions for the
potential energy can be used to establish the distance dependence of the
strength of the electric field generated by a dipole. We shall need this
expression when we calculate the dipole moment induced in one molecule by
another.
The starting point for the calculation is the strength of the electric field
generated by a point electric charge:
(18.20)
The field generated by a dipole is the sum of the fields generated by each
partial charge. For the point-dipole arrangement shown in Fig. 18.5,
Fig. 18.5 The electric field of a dipole is the sum of the opposing fields from
2
the positive and negative charges, each of which is proportional to 1/r . The
3
difference, the net field, is proportional to 1/r .
The same procedure that was used to derive the potential energy gives
(18.21)
The electric field of a multipole (in this case a dipole) decreases more rapidly
3
with distance (as 1/r for a dipole) than a monopole (a point charge).
19
(c) Dipole–dipole interactions
Comment 18.9
[The average (or mean value) of a function f(x) over the range from x = a to
x = b is
The volume element in polar coordinates is proportional to sin _ d_, and _
2
ranges from 0 to #. Therefore the average value of (1 _ 3 cos _) is
The potential energy of interaction between two
polar molecules is a complicated function of their
relative orientation. When the two dipoles are
parallel (as in 13), the potential energy is simply
(see Further information 18.1, derivation given
below)
(18.22)
This expression applies to polar molecules in a fixed, parallel, orientation in a
solid.
Further Information 18.1 The dipole–dipole interaction
An important problem in physical chemistry is the calculation of the potential
energy of interaction between two point dipoles with moments _1 and
_2,separated by a vector r. From classical electromagnetic theory, the
potential energy of _2 in the electric field
dot (scalar) product
1
generated by _1 is given by the
(18.46)
To calculate
1,we
consider a distribution of point charges qi located at xi,yi,
20
and zi from the origin. The Coulomb potential _ due to this distribution at a
point with coordinates x, y, and z is:
(18.47)
Comment 18.10
The potential energy of a charge q1 in the presence of another charge q2
may be written as V = q1_, where _ = q2/4#_0r is the Coulomb potential. If
there are several charges q2, q3, . . . .present in the system, then the total
potential experienced by the charge q1 is the sum of the potential generated
by each charge: _ = _2 + _3 + · · ·. The electric field strength is the
negative gradient of the electric potential: = ___. See Appendix 3 for more
details.
where r is the location of the point of interest and the ri are the locations of
the charges qi. If we suppose that all the charges are close to the origin (in
the sense that ri r),we can use a Taylor expansion to write where the ellipses
include the terms arising from derivatives with respect to yi and zi and
higher derivatives. If the charge distribution is electrically neutral,the first
term disappears because _iqi = 0. Next we note that ∑iqixi = µx,and likewise
for the y- and z-components. That is,
(18.49)
The electric field strength is (see Comment 18.10)
(18.50)
It follows from eqn 18.46 and eqn 18.50 that
(18.51)
21
For the arrangement shown in (13), in which m1·rµ1r cos _ and _2·r = µ2r
cos _, eqn 18.51 becomes:
(18.52)
which is eqn 18.22.
In a fluid of freely rotating molecules, the interaction between dipoles
averages to zero because f(_) changes sign as the orientation changes, and
its average value is zero. Physically, the like partial charges of two freely
rotating molecules are close together as much as the two opposite charges,
and the repulsion of the former is cancelled by the attraction of the latter.
The interaction energy of two freely rotating dipoles is zero. However,
because their mutual potential energy depends on their relative orientation,
the molecules do not in fact rotate completely freely, even in a gas. In fact,
the lower energy orientations are marginally favoured, so there is a nonzero
average interaction between polar molecules. We show in the following
Justification that the average potential energy of two rotating molecules that
are separated by a distance r is
(18.23)
This expression describes the Keesom interaction, and is the first of the
contributions to the van der Waals interaction.
Justification 18.5 The Keesom interaction
The detailed calculation of the Keesom interaction energy is quite
complicated, but the form of the final answer can be constructed quite
simply. First, we note that the average interaction energy of two polar
molecules rotating at a fixed separation r is given by
where f now includes a weighting factor in the averaging that is equal to the
probability that a particular orientation will be adopted. This probability is
_ /
given by the Boltzmann distribution p _e E kT, with E interpreted as the
potential energy of interaction of the two dipoles in that orientation. That is,
22
When the potential energy of interaction of the two dipoles is very small
compared with the energy of thermal motion, we can use V kT, expand the
exponential function in p, and retain only the first two terms:
The weighted average of f is therefore
where ···0 denotes an unweighted spherical average. The spherical average
2
of f is zero,so the first term vanishes. However, the average value of f is
2
nonzero because f is positive at all orientations, so we can write
2
2
The average value f 0 turns out to be /3 when the calculation is carried
through in detail. The final result is that quoted in eqn 18.23.
The important features of eqn 18.23 are its negative sign (the average
interaction is attractive), the dependence of the average interaction energy
on the inverse sixth power of the separation (which identifies it as a van der
Waals interaction), and its inverse dependence on the temperature. The last
feature reflects the way that the greater thermal motion overcomes the
mutual orientating effects of the dipoles at higher temperatures. The inverse
sixth power arises from the inverse third power of the interaction potential
energy that is weighted by the energy in the Boltzmann term, which is also
proportional to the inverse third power of the separation.
At 25°C the average interaction energy for pairs of molecules with µ = 1 D is
_1
about _0.07 kJ mol
when the separation is 0.5 nm. This energy should be
3
_1
compared with the average molar kinetic energy of /2RT = 3.7 kJ mol at
the same temperature. The interaction energy is also much smaller than the
energies involved in the making and breaking of chemical bonds.
23
(d) Dipole–induced-dipole interactions
Fig. 18.6 (a) A polar molecule (purple
arrow) can induce a dipole (white arrow) in
a nonpolar molecule, and (b) the latter ’s
orientation follows the former’s, so the
interaction does not average to zero.
A polar molecule with dipole moment µ1 can
induce a dipole µ2* in a neighbouring
polarizable molecule (Fig. 18.6). The
induced dipole interacts with the permanent
dipole of the first molecule,and the two are
attracted together. The average interaction
energy when the separation of the molecules is r is (for a derivation, see
Further reading)
(18.24)
where
2_ is the polarizability volume of molecule 2 and µ1 is the
permanent dipole moment of molecule 1. Note that the C in this expression
is different from the C in eqn 18.23 and other expressions below: we are
6
using the same symbol in C/r to emphasize the similarity of form of each
expression.
The dipole–induced-dipole interaction energy is independent of the
temperature because thermal motion has no effect on the averaging
process. Moreover, like the dipole–dipole interaction, the potential energy
6
3
depends on 1/r : this distance dependence stems from the 1/r dependence
3
of the field (and hence the magnitude of the induced dipole) and the 1/r
dependence of the potential energy of interaction between the permanent
and induced dipoles. For a molecule with µ = 1 D (such as HCl) near a
molecule of polarizability volume
_ = 10 _ 10
_30
3
m (such as benzene,
_1
Table 18-1), the average interaction energy is about _0.8 kJ mol
the separation is 0.3 nm.
when
24
(e) Induced-dipole–induced-dipole interactions
Nonpolar molecules (including closed-shell atoms, such as Ar) attract one
another even though neither has a permanent dipole moment. The abundant
evidence for the existence of interactions between them is the formation of
condensed phases of nonpolar substances, such as the condensation of
hydrogen or argon to a liquid at low temperatures and the fact that benzene
is a liquid at normal temperatures.
Fig. 18.7 (a) In the dispersion interaction,
an instantaneous dipole on one molecule
induces a dipole on another molecule, and
the two dipoles then interact to lower the
energy. (b) The two instantaneous dipoles
are correlated and, although they occur in
different orientations at different instants,
the interaction does not average to zero.
The interaction between nonpolar
molecules arises from the transient dipoles
that all molecules possess as a result of
fluctuations in the instantaneous positions
of electrons. To appreciate the origin of the
interaction, suppose that the electrons in
one molecule flicker into an arrangement that gives the molecule an
instantaneous dipole moment µ1*. This dipole generates an electric field that
polarizes the other molecule,and induces in that molecule an instantaneous
dipole moment µ2*. The two dipoles attract each other and the potential
energy of the pair is lowered. Although the first molecule will go on to
change the size and direction of its instantaneous dipole, the electron
distribution of the second molecule will follow; that is, the two dipoles are
correlated in direction (Fig. 18.7). Because of this correlation, the attraction
between the two instantaneous dipoles does not average to zero, and gives
rise to an induced-dipole–induced-dipole interaction. This interaction is
called either the dispersion interaction or the London interaction (for
Fritz London, who first described it).
Polar molecules also interact by a dispersion interaction: such molecules also
possess instantaneous dipoles, the only difference being that the time
average of each fluctuating dipole does not vanish, but corresponds to the
permanent dipole. Such molecules therefore interact both through their
25
permanent dipoles and through the correlated, instantaneous fluctuations in
these dipoles.
The strength of the dispersion interaction depends on the polarizability of the
first molecule because the instantaneous dipole moment µ1* depends on the
looseness of the control that the nuclear charge exercises over the outer
electrons. The strength of the interaction also depends on the polarizability
of the second molecule,for that polarizability determines how readily a dipole
can be induced by another molecule. The actual calculation of the dispersion
interaction is quite involved, but a reasonable approximation to the
interaction energy is given by the London formula:
(18.25)
where I1 and I2 are the ionization energies of the two molecules (Table 104). This interaction energy is also proportional to the inverse sixth power of
the separation of the molecules, which identifies it as a third contribution to
the van der Waals interaction. The dispersion interaction generally
dominates all the interactions between molecules other than hydrogen
bonds.
Illustration 18.1 Calculating the strength of the dispersion interaction
For two CH4 molecules, we can substitute _ = 2.6 _ 10
mol
_1
_1
to obtain V = _2 kJ mol
_30
3
m and I ≈ 700 kJ
for r = 0.3 nm. A very rough check on this
_1
figure is the enthalpy of vaporization of methane, which is 8.2 kJ mol .
However, this comparison is insecure, partly because the enthalpy of
vaporization is a many-body quantity and partly because the long-distance
assumption breaks down.
26
(f) Hydrogen bonding
The interactions described so far are universal in the sense that they are
possessed by all molecules independent of their specific identity. However,
there is a type of interaction possessed by molecules that have a particular
constitution. A hydrogen bond is an attractive interaction between two
species that arises from a link of the form A-H···B, where A and B are highly
electronegative elements and B possesses a lone pair of electrons. Hydrogen
bonding is conventionally regarded as being limited to N, O, and F but, if B is
an anionic species (such as Cl_), it may also participate in hydrogen
bonding. There is no strict cutoff for an ability to participate in hydrogen
bonding, but N, O, and F participate most effectively.
Fig. 18.8 The molecular orbital
interpretation of the formation of an
A—H···B hydrogen bond. From the three A,
H, and B orbitals, three molecular orbitals
can be formed (their relative contributions
are represented by the sizes of the
spheres. Only the two lower energy orbitals
are occupied, and there may therefore be a
net lowering of energy compared with the
separate AH and B species.
The formation of a hydrogen bond can be
regarded either as the approach between a
partial positive charge of H and a partial
negative charge of B or as a particular
example of delocalized molecular orbital
formation in which A, H, and B each supply
one atomic orbital from which three
molecular orbitals are constructed
(Fig. 18.8). Thus, if the A-H bond is
regarded as formed from the overlap of an orbital on A, _A,and a hydrogen
1s orbital, _H, and the lone pair on B occupies an orbital on B, _B, then,
when the two molecules are close together, we can build three molecular
orbitals from the three basis orbitals:
One of the molecular orbitals is bonding, one almost nonbonding, and the
third antibonding. These three orbitals need to accommodate four electrons
27
(two from the original A-H bond and two from the lone pair of B), so two
enter the bonding orbital and two enter the nonbonding orbital. Because the
antibonding orbital remains empty, the net effect—depending on the precise
location of the almost nonbonding orbital—may be a lowering of energy.
_1
In practice, the strength of the bond is found to be about 20 kJ mol .
Because the bonding depends on orbital overlap, it is virtually a contact-like
interaction that is turned on when AH touches B and is zero as soon as the
contact is broken. If hydrogen bonding is present, it dominates the other
intermolecular interactions. The properties of liquid and solid water, for
example, are dominated by the hydrogen bonding between H2O molecules.
The structure of DNA and hence the transmission of genetic information is
crucially dependent on the strength of hydrogen bonds between base pairs.
The structural evidence for hydrogen bonding comes from noting that the
internuclear distance between formally non-bonded atoms is less than their
van der Waals contact distance, which suggests that a dominating attractive
interaction is present. For example, the O-O distance in O-H···O is expected
to be 280 pm on the basis of van der Waals radii, but is found to be 270 pm
in typical compounds. Moreover, the H···O distance is expected to be 260
pm but is found to be only 170 pm.
Hydrogen bonds may be either symmetric or unsymmetric. In a symmetric
hydrogen bond, the H atoms lies midway between the two other atoms. This
_
arrangement is rare, but occurs in F—H···F , where both bond lengths are
120 pm. More common is the unsymmetrical arrangement, where the A—H
bond is shorter than the H···B bond. Simple electrostatic arguments, treating
A—H···B as an array of point charges (partial negative charges on A and B,
partial positive on H) suggest that the lowest energy is achieved when the
bond is linear, because then the two partial negative charges are furthest
apart. The experimental evidence from structural studies support a linear or
near-linear arrangement.
28
(g) The hydrophobic interaction
Fig. 18.9 When a hydrocarbon molecule is
surrounded by water, the H2O molecules
form a clathrate cage. As a result of this
acquisition of structure, the entropy of the
water decreases, so the dispersal of the
hydrocarbon into the water is entropyopposed; its coalescence is entropyfavoured.
Nonpolar molecules do dissolve slightly in polar solvents, but strong
interactions between solute and solvent are not possible and as a result it is
found that each individual solute molecule is surrounded by a solvent cage
(Fig. 18.9). To understand the consequences of this effect, consider the
thermodynamics of transfer of a nonpolar hydrocarbon solute from a
nonpolar solvent to water, a polar solvent. Experiments indicate that the
process is endergonic (∆transferG > 0),as expected on the basis of the
increase in polarity of the solvent, but exothermic (∆transferH < 0). Therefore,
it is a large decrease in the entropy of the system (∆transferS < 0) that
accounts for the positive Gibbs energy of transfer. For example,the process
_1
_1
has ∆transferG = +12 kJ mol , ∆transferH = _10 kJ mol ,and ∆transferS = _75 J
_1
_1
K mol at 298 K. Substances characterized by a positive Gibbs energy of
transfer from a nonpolar to a polar solvent are called hydrophobic.
It is possible to quantify the hydrophobicity of a small molecular group R by
defining the hydrophobicity constant, ", as
(18.26)
where S is the ratio of the molar solubility of the compound R-A in octanol, a
nonpolar solvent, to that in water, and S0 is the ratio of the molar solubility
of the compound H-A in octanol to that in water. Therefore,positive values of
29
" indicate hydrophobicity and negative values of " indicate hydrophilicity,
the thermodynamic preference for water as a solvent. It is observed
experimentally that the " values of most groups do not depend on the
nature of A. However, measurements do suggest group additivity of "
values. For example, " for R = CH3, CH2CH3, (CH2)2CH3, (CH2)3CH3,and
(CH2)4CH3 is, respectively, 0.5, 1.0, 1.5, 2.0, and 2.5 and we conclude that
acyclic saturated hydrocarbons become more hydrophobic as the carbon
chain length increases. This trend can be rationalized by ∆transferH becoming
more positive and ∆transferS more negative as the number of carbon atoms in
the chain increases.
At the molecular level,formation of a solvent cage around a hydrophobic
molecule involves the formation of new hydrogen bonds among solvent
molecules. This is an exothermic process and accounts for the negative
values of ∆transferH. On the other hand, the increase in order associated with
formation of a very large number of small solvent cages decreases the
entropy of the system and accounts for the negative values of ∆transferS.
However, when many solute molecules cluster together, fewer (albeit larger)
cages are required and more solvent molecules are free to move. The net
effect of formation of large clusters of hydrophobic molecules is then a
decrease in the organization of the solvent and therefore a net increase in
entropy of the system. This increase in entropy of the solvent is large
enough to render spontaneous the association of hydrophobic molecules in a
polar solvent.
The increase in entropy that results from fewer structural demands on the
solvent placed by the clustering of nonpolar molecules is the origin of the
hydrophobic interaction, which tends to stabilize groupings of
hydrophobic groups in micelles and biopolymers (Chapter 19). The
hydrophobic interaction is an example of an ordering process that is
stabilized by a tendency toward greater disorder of the solvent.
30
(h) The total attractive interaction
We shall consider molecules that are unable to participate in hydrogen bond
formation. The total attractive interaction energy between rotating molecules
is then the sum of the three van der Waals contributions discussed above.
(Only the dispersion interaction contributes if both molecules are nonpolar.)
In a fluid phase, all three contributions to the potential energy vary as the
inverse sixth power of the separation of the molecules, so we may write
(18.27)
where C6 is a coefficient that depends on the identity of the molecules.
Although attractive interactions between molecules are often expressed as in
eqn 18.27, we must remember that this equation has only limited validity.
First, we have taken into account only dipolar interactions of various kinds,
for they have the longest range and are dominant if the average separation
of the molecules is large. However, in a complete treatment we should also
consider quadrupolar and higher-order multipole interactions, particularly if
the molecules do not have permanent electric dipole moments. Secondly,
the expressions have been derived by assuming that the molecules can
rotate reasonably freely. That is not the case in most solids, and in rigid
3
media the dipole–dipole interaction is proportional to 1/r because the
Boltzmann averaging procedure is irrelevant when the molecules are trapped
into a fixed orientation.
A different kind of limitation is that eqn 18.27 relates to the interactions of
pairs of molecules. There is no reason to suppose that the energy of
interaction of three (or more) molecules is the sum of the pairwise
interaction energies alone. The total dispersion energy of three closed-shell
atoms, for instance, is given approximately by the Axilrod–Teller formula:
(18.28a)
where
(18.28b)
31
The parameter a is approximately equal to
3
/4 _C6; the angles _ are the internal angles
of the triangle formed by the three atoms
(14). The term in C_ (which represents the
non-additivity of the pairwise interactions) is
negative for a linear arrangement of atoms (so
that arrangement is stabilized) and positive for
an equilateral triangular cluster. It is found
that the three-body term contributes about 10
per cent of the total interaction energy in
liquid argon.
32
18.5 Repulsive and total interactions
When molecules are squeezed together, the nuclear and electronic
repulsions and the rising electronic kinetic energy begin to dominate the
attractive forces. The repulsions increase steeply with decreasing separation
in a way that can be deduced only by very extensive, complicated molecular
structure calculations of the kind described in Chapter 11 (Fig. 18.10).
Fig. 18.10 The general form of
an intermolecular potential
energy curve. At long range the
interaction is attractive, but at
close range the repulsions
dominate.
Fig. 18.11 The Lennard-Jones potential, and
the relation of the parameters to the features
of the curve. The green and purple lines are the
two contributions.
In many cases, however, progress can be made by using a greatly simplified
representation of the potential energy, where the details are ignored and the
general features expressed by a few adjustable parameters. One such
approximation is the hard-sphere potential, in which it is assumed that
the potential energy rises abruptly to infinity as soon as the particles come
within a separation d:
33
(18.29)
This very simple potential is surprisingly useful for assessing a number of
properties. Another widely used approximation is the Mie potential:
(18.30)
with n > m. The first term represents repulsions and the second term
attractions. The Lennard-Jones potential is a special case of the Mie
potential with n = 12 and m = 6 (Fig. 18.11); it is often written in the form
(18.31)
The two parameters are _, the depth of the well (not to be confused with the
symbol of the permittivity of a medium used in Section 18-3), and r0,the
separation at which V = 0 (Table 18-4). The well minimum occurs at re =
2
1/6
r0. Although the Lennard-Jones potential has been used in many
12
calculations,there is plenty of evidence to show that 1/r is a very poor
representation of the repulsive potential, and that an exponential form,
_ /
e r r0, is greatly superior. An exponential function is more faithful to the
exponential decay of atomic wavefunctions at large distances, and hence to
the overlap that is responsible for repulsion. The potential with an
6
exponential repulsive term and a 1/r attractive term is known as an exp-6
potential. These potentials can be used to calculate the virial coefficients of
gases, as explained in Section 17-5, and through them various properties
of real gases, such as the Joule–Thompson coefficient. The potentials are
also used to model the structures of condensed fluids.
34
With the advent of atomic force microscopy (AFM), in which the force
between a molecular sized probe and a surface is monitored (see
Impact I9.1), it has become possible to measure directly the forces acting
between molecules. The force, F, is the negative slope of potential, so for a
Lennard-Jones potential between individual molecules we write
(18.32)
26
The net attractive force is greatest (from dF/dr = 0) at r = ( /7)
7
7/6
1/6
r0,or
1.244r0, and at that distance is equal to _144( /26) _/13r0,or _2.397_/r0.
For typical parameters, the magnitude of this force is about 10 pN.
35
IMPACT ON MEDICINE_I.18.1
Molecular recognition and drug design
A drug is a small molecule or protein that binds to a specific receptor site of
a target molecule, such as a larger protein or nucleic acid, and inhibits the
progress of disease. To devise efficient therapies, we need to know how to
characterize and optimize molecular interactions between drug and target.
Molecular interactions are responsible for the assembly of many biological
structures. Hydrogen bonding and hydrophobic interactions are primarily
responsible for the three-dimensional structures of biopolymers, such as
proteins, nucleic acids, and cell membranes. The binding of a ligand, or
guest, to a biopolymer, or host, is also governed by molecular interactions.
Examples of biological host–guest complexes include enzyme–substrate
complexes, antigen–antibody complexes, and drug–receptor complexes. In
all these cases, a site on the guest contains functional groups that can
interact with complementary functional groups of the host. For example, a
hydrogen bond donor group of the guest must be positioned near a
hydrogen bond acceptor group of the host for tight binding to occur. It is
generally true that many specific intermolecular contacts must be made in a
biological host–guest complex and, as a result, a guest binds only hosts that
are chemically similar. The strict rules governing molecular recognition of a
guest by a host control every biological process, from metabolism to
immunological response, and provide important clues for the design of
effective drugs for the treatment of disease.
Fig. 18.12 Some drugs with
planar π systems, shown as a
green rectangle, intercalate
between base pairs of DNA.
Interactions between nonpolar
groups can be important in the
binding of a guest to a host. For
example, many enzyme active
sites have hydrophobic pockets
that bind nonpolar groups of a
substrate. In addition to dispersion, repulsive, and hydrophobic interactions,
" stacking interactions are also possible, in which the planar " systems of
aromatic macrocycles lie one on top of the other, in a nearly parallel
36
orientation. Such interactions are responsible for the stacking of hydrogenbonded base pairs in DNA (Fig. 18.12). Some drugs with planar " systems,
shown as a green rectangle in Fig. 18.12, are effective because they
intercalate between base pairs through " stacking interactions, causing the
helix to unwind slightly and altering the function of DNA.
Coulombic interactions can be important in the interior of a biopolymer host,
where the relative permittivity can be much lower than that of the aqueous
exterior. For example, at physiological pH, amino acid side chains containing
carboxylic acid or amine groups are negatively and positively charged,
respectively, and can attract each other. Dipole–dipole interactions are also
possible because many of the building blocks of biopolymers are polar,
including the peptide link, —CONH— (see Example 18.1). However,
hydrogen bonding interactions are by far the most prevalent in a biological
host–guest complexes. Many effective drugs bind tightly and inhibit the
action of enzymes that are associated with the progress of a disease. In
many cases, a successful inhibitor will be able to form the same hydrogen
bonds with the binding site that the normal substrate of the enzyme can
form, except that the drug is chemically inert toward the enzyme.
There are two main strategies for the discovery of a drug. In structure-based
design, new drugs are developed on the basis of the known structure of the
receptor site of a known target. However, in many cases a number of socalled lead compounds are known to have some biological activity but little
information is available about the target. To design a molecule with
improved pharmacological efficacy, quantitative structure–activity
relationships (QSAR) are often established by correlating data on activity
of lead compounds with molecular properties, also called molecular
descriptors, which can be determined either experimentally or
computationally.
In broad terms, the first stage of the QSAR method consists of compiling
molecular descriptors for a very large number of lead compounds.
Descriptors such as molar mass, molecular dimensions and volume, and
relative solubility in water and nonpolar solvents are available from routine
experimental procedures. Quantum mechanical descriptors determined by
semi-empirical and ab initio calculations include bond orders and HOMO and
LUMO energies.
In the second stage of the process, biological activity is expressed as a
function of the molecular descriptors. An example of a QSAR equation is:
37
(18.33)
where di is the value of the descriptor and ci is a coefficient calculated by
fitting the data by regression analysis. The quadratic terms account for the
fact that biological activity can have a maximum or minimum value at a
specific descriptor value. For example,a molecule might not cross a biological
membrane and become available for binding to targets in the interior of the
cell if it is too hydrophilic (water-loving), in which case it will not partition
into the hydrophobic layer of the cell membrane (see Section 19-14 for
details of membrane structure), or too hydrophobic (water-repelling), for
then it may bind too tightly to the membrane. It follows that the activity will
peak at some intermediate value of a parameter that measures the relative
solubility of the drug in water and organic solvents.
In the final stage of the QSAR process, the activity of a drug candidate can
be estimated from its molecular descriptors and the QSAR equation either by
interpolation or extrapolation of the data. The predictions are more reliable
when a large number of lead compounds and molecular descriptors are used
to generate the QSAR equation.
The traditional QSAR technique has been refined into 3D QSAR, in which
sophisticated computational methods are used to gain further insight into
the three-dimensional features of drug candidates that lead to tight binding
to the receptor site of a target. The process begins by using a computer to
superimpose three-dimensional structural models of lead compounds and
looking for common features, such as similarities in shape, location of
functional groups, and electrostatic potential plots, which can be obtained
from molecular orbital calculations. The key assumption of the method is
that common structural features are indicative of molecular properties that
enhance binding of the drug to the receptor. The collection of superimposed
molecules is then placed inside a three-dimensional grid of points. An atomic
3
probe, typically an sp -hybridized carbon atom, visits each grid point and
two energies of interaction are calculated: Esteric, the steric energy reflecting
interactions between the probe and electrons in uncharged regions of the
drug, and Eelec,the electrostatic energy arising from interactions between the
probe and a region of the molecule carrying a partial charge. The measured
equilibrium constant for binding of the drug to the target, Kbind, is then
assumed to be related to the interaction energies at each point r by the 3D
QSAR equation
38
(18.34)
Fig. 18.13 A 3D QSAR analysis of the
binding of steroids, molecules with the
carbon skeleton shown, to human
corticosteroid-binding globulin (CBG).
The ellipses indicate areas in the protein
’s binding site with positive or negative
electrostatic potentials and with little or
much steric crowding. It follows from
the calculations that addition of large
substituents near the left-hand side of
the molecule (as it is drawn on the
page) leads to poor affinity of the drug
to the binding site. Also, substituents
that lead to the accumulation of
negative electrostatic potential at either
end of the drug are likely to show
enhanced affinity for the binding site.
(Adapted from P. Krogsgaard-Larsen, T.
Liljefors, U. Madsen (ed.), Textbook of
drug design and discovery, Taylor &
Francis, London (2002).)
where the c(r) are coefficients calculated by regression analysis, with the
coefficients cS and cE reflecting the relative importance of steric and
electrostatic interactions, respectively, at the grid point r. Visualization of
the regression analysis is facilitated by colouring each grid point according to
the magnitude of the coefficients. Figure 18.13 shows results of a 3D QSAR
analysis of the binding of steroids, molecules with the carbon skeleton
shown, to human corticosteroid-binding globulin (CBG). Indeed, we see that
the technique lives up to the promise of opening a window into the chemical
nature of the binding site even when its structure is not known.
The QSAR and 3D QSAR methods, though powerful, have limited power: the
predictions are only as good as the data used in the correlations are both
reliable and abundant. However, the techniques have been used successfully
to identify compounds that deserve further synthetic elaboration, such as
addition or removal of functional groups, and testing.
........................................
Further Reading
39
Articles and texts
P.W. Atkins and R.S. Friedman, Molecular quantum mechanics. Oxford
University Press (2005).
M.A.D. Fluendy and K.P. Lawley, Chemical applications of molecular beam
scattering. Chapman and Hall, London (1973).
-->J.N. Israelachvili, Intermolecular and surface forces. Academic
Press, New York (1998). [A great book! Now I know why everyone
refers to this book.]
-->G.A. Jeffrey, An introduction to hydrogen bonding. Oxford
University Press (1997).
H.-J. Schneider and A. Yatsimirsky, Principles and methods in
supramolecular chemistry. Wiley, Chichester (1999).
Sources of data and information
J.J. Jasper, The surface tension of pure liquid compounds. J. Phys. Chem.
Ref. Data 1, 841 (1972).
D.R. Lide (ed.), CRC Handbook of Chemistry and Physics, Sections 3, 4, 6, 9,
10, 12, and 13. CRC Press, Boca Raton (2000).
..........................
Discussion Questions
18.4 Account for the theoretical conclusion that many attractive
6
interactions between molecules vary with their separation as 1/r .
18.5 Describe the formation of a hydrogen bond in terms of molecular
orbitals.
18.6 Account for the hydrophobic interaction and discuss its manifestations.
Exercises
18.1a Which of the following molecules may be polar: CIF3, O3, H2O2?
18.2a The electric dipole moment of toluene (methylbenzene) is 0.4 D.
Estimate the dipole moments of the three xylenes (dimethylbenzene). Which
answer can you be sure about?
18.2b Calculate the resultant of two dipole moments of magnitude 1.5 D
and 0.80 D that make an angle of 109.5° to each other.
Correct Answer µ = 1.4 D.
18.3a Calculate the magnitude and direction of the dipole moment of the
following arrangement of charges in the xy-plane: 3e at (0,0), _e at (0.32
nm, 0), and _2e at an angle of 20° from the x-axis and a distance of 0.23
40
nm from the origin.
18.3b Calculate the magnitude and direction of the dipole moment of the
following arrangement of charges in the xy-plane: 4e at (0, 0), _2e at (162
pm, 0), and _2e at an angle of 30° from the x-axis and a distance of 143 pm
from the origin.
Correct Answer
µ = 9.45 _ 10_29 C m, _ = 194.0°.
Numerical problems
18.1 Suppose an H2O molecule (µ = 1.85 D) approaches an anion. What is
the favourable orientation of the molecule? Calculate the electric field (in
volts per metre) experienced by the anion when the water dipole is (a) 1.0
nm, (b) 0.3 nm, (c) 30 nm from the ion.
Correct Answer
SF4.
18.2 An H2O molecule is aligned by an external electric field of strength 1.0
_1
_24
3
kV m and an Ar atom (_ = 1.66 _ 10
cm ) is brought up slowly from one
side. At what separation is it energetically favourable for the H2O molecule
to flip over and point towards the approaching Ar atom?
Correct Answer
µ = 1.4 D.
18.3 The relative permittivity of chloroform was measured over a range of
temperatures with the following results:
The freezing point of chloroform is _64°C. Account for these results and
calculate the dipole moment and polarizability volume of the molecule.
Correct Answer µ = 9.45 _ 10_29 C m, _ = 194.0°.
18.4 The relative permittivities of methanol (m.p. –95°C) corrected for
density variation are given below. What molecular information can be
deduced from these values? Take _ = 0.791 g cm
–3
at 20°C.
Correct Answer µ = 3.23 _ 10_30 C m, _ = 2.55 _ 10_39 C2 m2 J_1.
18.5 In his classic book Polar molecules, Debye reports some early
measurements of the polarizability of ammonia. From the selection below,
determine the dipole moment and the polarizability volume of the molecule.
The refractive index of ammonia at 273 K and 100 kPa is 1.000 379 (for
yellow sodium light). Calculate the molar polarizability of the gas at this
temperature and at 292.2 K. Combine the value calculated with the static
molar polarizability at 292.2 K and deduce from this information alone the
molecular dipole moment.
Correct Answer
_r = 8.97.
18.13 Show that the mean interaction energy of N atoms of diameter d
41
6
interacting with a potential energy of the form C6/R is given by U =
2
3
_2N C6/3Vd ,where V is the volume in which the molecules are confined and
all effects of clustering are ignored. Hence, find a connection between the
2
2
van der Waals parameter a and C6, from n alV = (∂U/∂V)T.
Correct Answer
(a) CH2Cl2; (b) CH3CH3; (d) N2O.
18.14 Suppose the repulsive term in a Lennard-Jones (12,6)-potential is
/
replaced by an exponential function of the form e_r d. Sketch the form of
the potential energy and locate the distance at which it is a minimum.
Correct Answer
K ≈ 0.25.
18.15 The cohesive energy density,, is defined as U/V, where U is the mean
potential energy of attraction within the sample and V its volume. Show that
1
= /2∫V(R)d_, where is the number density of the molecules and V(R) is
their attractive potential energy and where the integration ranges from d to
infinity and over all angles. Go on to show that the cohesive energy density
of a uniform distribution of molecules that interact by a van der Waals
6
2
3
2
2
attraction of the form _C6/R is equal to (2"/3)(NA /d M )_ C6,where _ is the
mass density of the solid sample and M is the molar mass of the molecules.
Correct Answer
B1 = 9.40 _ 10_4 T, 6.25 µs.
Applications: to biochemistry
18.18 Phenylalanine (Phe, 15) is a naturally occurring amino acid. What is
the energy of interaction between its phenyl group and the electric dipole
moment of a neighbouring peptide group?
Take the distance between the groups as 4.0 nm and treat the phenyl group
as a benzene molecule. The dipole moment of the peptide group is µ = 2.7 D
_29
3
and the polarizability volume of benzene is _ = 1.04 _ 10
m .
Correct Answer
2.2 mT, g = 1.992.
18.19 Now consider the London interaction between the phenyl groups of
two Phe residues (see Problem 18.18). (a) Estimate the potential energy of
42
interaction between two such rings (treated as benzene molecules)
separated by 4.0 nm. For the ionization energy, use I = 5.0 eV. (b) Given
that force is the negative slope of the potential, calculate the distance
dependence of the force acting between two nonbonded groups of atoms,
such as the phenyl groups of Phe, in a polypeptide chain that can have a
London dispersion interaction with each other. What is the separation at
which the force between the phenyl groups (treated as benzene molecules)
of two Phe residues is zero? Hint. Calculate the slope by considering the
potential energy at r and r + _r, with _r r, and evaluating {V(r + _r) _
V(r)}/_r. At the end of the calculation, let _r become vanishingly small.
Correct Answer
Eight equal parts at ±1.445 ± 1.435 ± 1.055 mT from the centre, namely:
328.865, 330.975, 331.735, 331.755, 333.845, 333.865, 334.625 and
336.735 mT.
18.20 Molecular orbital calculations may be used to predict structures of
intermolecular complexes. Hydrogen bonds between purine and pyrimidine
bases are responsible for the double helix structure of DNA (see Chapter
19). Consider methyl-adenine (16, with R = CH3) and methyl-thymine (17,
with R = CH3) as models of two bases that can form hydrogen bonds in DNA.
(a) Using molecular modelling software and the computational method of
your choice, calculate the atomic charges of all atoms in methyl-adenine and
methyl-thymine. (b) Based on your tabulation of atomic charges, identify the
atoms in methyl-adenine and methyl-thymine that are likely to participate in
hydrogen bonds. (c) Draw all possible adenine–thymine pairs that can be
linked by hydrogen bonds, keeping in mind that linear arrangements of the
A-H···B fragments are preferred in DNA. For this step, you may want to use
your molecular modelling software to align the molecules properly. (d)
Consult Chapter 19 and determine which of the pairs that you drew in part
(c) occur naturally in DNA molecules. (e) Repeat parts (a)–(d) for cytosine
and guanine, which also form base pairs in DNA (see Chapter 19 for the
structures of these bases).
Correct Answer D0 = 3.235 _ 104 cm_1 = 4.01 eV.
43
18.21 Molecular orbital calculations may be used to predict the dipole
moments of molecules. (a) Using molecular modelling software and the
computational method of your choice, calculate the dipole moment of the
peptide link, modelled as a trans-N-methylacetamide (18).
Plot the energy of interaction between these dipoles against the angle _ for r
= 3.0 nm (see eqn 18.22). (b) Compare the maximum value of the
_1
dipole–dipole interaction energy from part (a) to 20 kJ mol , a typical value
for the energy of a hydrogen-bonding interaction in biological systems.
Correct Answer (a) 332.3 mT; (b) 1209 mT.